For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers.

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Magnitude of rotation between two quaternions

I have a quaternion for an object's starting rotation, and a quaternion for an object's ending rotation, and I am SLERPing the shortest rotation between the two. How can I figure out the magnitude of ...
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61 views

Every element with finite conjugates in the ring of real quaternions is a real number

Let $H$ be the ring of real quaternions and let $x$ be a member of $H$ having finite conjugates. Prove that $x$ is a real number. I worked a lot on this question, but no progress! :|
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113 views

Relative rotation between quaternions

Say I have a quaternion q which describes how to get from frame 0 to frame 1, and a quaternion r which describes how to get from frame 0 to frame 2. To get the "quaternion difference" between q and r, ...
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190 views

Using quaternions to represent an affine transformation?

I have never used quaternions, so before trying on my problem I would like to know whether this is a good idea: I want to interpolate an affine transformation: I have a set of points in a first 2D ...
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76 views

Show that for $\forall a\in\mathbb{H}, \ \exists b \in\mathbb{H}: ab =ba = 1$.

Show that $\forall a\in\mathbb{H}, \ \exists b \in\mathbb{H}: ab =ba = 1.$ I am pretty sure I can easily google the multiplicative inverse in $\mathbb{H}$, but can you give me a hint on how to ...
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195 views

Difference between quaternions and rotation matrices

This is a really simple question, I guess. Do quaternions cover the same set of rotations as rotation matrices? I assume the answer is yes, they both can represent SO(3), but I'm unsure about the ...
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1answer
28 views

Enumeration of Hurwitz quaternions of norm p

In "on Quaternions and Octonions" by Conway and Smith, they quote a result by which for each prime norm $p$ there are exactly $p+1$ Hurwitz quaternions of norm $p$. I haven't found any proof of that. ...
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54 views

Hamilton's letter to his son

I'm looking for a better reference on this letter from Hamilton to his son where he wrote about his discovering on Quaternions. I'd like to read, if it is possible, a scanned version of the letter. ...
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43 views

Are there any methods to exponentiate a real number with a number from an arbitrary field?

How can I take the following exponent, for some real-valued number a? $$a^{3+2j-9k+3i}$$ over the field of quaternions, or any field for that matter? On wikipedia we are given the following formula, ...
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78 views

Why do we start losing algebraic properties when dealing with hypercomplex numbers? [duplicate]

Every form of hypercomplex number I have seen (including the complex numbers) lose some important algebraic property. Why is that? Is there a pattern to what we lose?
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47 views

How stable is my quaternion interpolation?

after some experimentation to optimize slerp I found that finding the middle between quaternion is rather cheap (for $t=0.5$) in particular: (with $\theta$ the angle between $q_1$ and $q_1$) ...
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234 views

Examples of a non commutative division ring

What are some examples of a non commutative division ring other than quaternions?
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63 views

Why and how are quaternions 'bilinear'?

What does it mean when we say that quaternion composition is 'bilinear'? I have observed that some authors write quaternion multiplication as: While others specify: Excuse the poor images, ...
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1answer
71 views

Show that the the multipliction and inverse operations on the quaternion unit sphere are continuous

This is a bit of a tricky question, we define the real Quaternions as: $$H=\left\{ a+bi+cj+dk\mid a,b,c,d\in\mathbb{R}\right\}$$ With the rule that: $$ij=-ji=k\:,\: jk=-kj=i\:,\: ki=-ik=\, j\;,\: ...
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389 views

Rotation by quaternion conjugation and quaternion matrix

A rotation of vector $v$ can be done by matrix multiplication $Q^{*}Qv$ where $Q=\begin{pmatrix}w & -z & y & x \\ z & w & -x & y \\ -y &x &w& z\\ -x& -y ...
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18 views

Reference for the polar parameterization of quaternions

I would like to find the original reference in which the polar parameterization of quaternions was given (i.e. the relationship between the components of a unit quaternion and the polar angles of an ...
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1answer
75 views

Automorphism of $Q_8$ [duplicate]

Is there anyone could help me to prove that $Aut(Q_8)=S_4$? Someone told me that there's an isomorphism between the rigid motions of cube and $Aut(Q_8)$, any ideas? Thank you!
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90 views

Show exponential function maps any line through the origin onto a circle of radius 1 in $\mathbb{S}^3$.

Show that the exponential function maps any line through the origin in $\mathbb{R}$i +$\mathbb{R}$j + $\mathbb{R}$k onto a circle of radius 1 in $\mathbb{S}^3$. I know that for any element v $\in$ ...
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How to rotate one vector about another?

Breif Having given 2 non-parallel vectors: a and b, is there any way by which I may rotate a about b such that b acts as the axis about which a is rotating. Question Given: vector a and b To find: ...
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135 views

Is it possible to use the imaginary components of quaternions to facilitate calculation of vector cross products?

It has come to my attention that the cross products of the vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are almost identical to the products of the imaginary components of quaternions $i$, ...
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1answer
37 views

Set of rotations necessary to connect two points in R³ using a thin cylinder

I have been scratching my head for days trying to answer this question. Suppose i have 2 points on three-dimensional space, say, $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, and they are separated by ...
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60 views

Present-day uses of quaternions [duplicate]

"Everybody knows" that quaternions are not used for the purposes for which they were originally intended. They are, however, used in computer graphics, and perhaps in astrogation. Besides that, what ...
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185 views

Understanding quaternions & gradient descent in a paper on inertial / magnetic sensor arrays

I hope this question is appropriate here! I and a friend at work are trying to understand Sebastian Madgwick's paper, "An efficient orientation for inertial and inertial/magnetic sensor arrays" ...
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146 views

Relationship between two maps from $SU(2)$ to $SO(3)$

I have two maps from $SU(2) \to SO(3)$. For the first map, think of $SU(2)$ as the group of unit quaternions. Under this identification, we can give a map $f: SU(2) \to SO(3)$ given by $SU(2)$ ...
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181 views

Why is $x_1 i = i x_1$ for quaternions?

According to Wikipedia, $$x+y = (x_0+y_0)+(x_1+y_1) i+(x_2+y_2) j+(x_3+y_3) k$$ and $$\begin{align} x y &=( x_0 y_0 - x_1 y_1 - x_2 y_2 - x_3 y_3)\\ &+( x_0 y_1 + x_1 y_0 + x_2 y_3 - x_3 ...
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118 views

Why is $(-1) \cdot j = j \cdot (-1)$ for quaternions?

I'm currently trying to understand the following part of a script (translated from German to English). It is the first part where quaternions get introduced, so I don't know anything about them except ...
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247 views

Trinonions, Quaternions, Quinonions, Sextonions, Septonions, Octonions

There are quaternions and octonions and even sextonions but what about trinonions, quinonions and septonions. Are there 3, 5, and 7 dimensional algebras which could be called trinonions, quinonions ...
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268 views

Small angular displacements with a quaternion representation

I have the orientation of a 3D spatial object represented by a unit quaternion: $$ q = a_1 + a_2 i + a_3 j + a_4 k $$ $$ \|q\| = (a_1^2 + a_2^2 + a_3^2 + a_4^2)^{1/2} = 1 $$ I'd like to perturb this ...
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267 views

Learning track for quaternions?

I've been out of the math loop for the last decade (although I'm a programmer, I've not done anything with calculus or above for this long) and I'd like to learn about quaternions (particularly ...
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61 views

Pfister's 16-Square Identity and the norm of sedenions

Consider the sequence of numbers: complex numbers $\Bbb C$, quaternions $\Bbb H$, octonions $\Bbb O$, and sedenions $\Bbb S$. The Brahmagupta-Fibonacci 2-Square identity implies that the norm of the ...
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233 views

3D Positioning Vectors to Matrix & Quaternions

This is also programming related, but I think it's more about math than anything else. I have an Object which I want to represent it's 3d position, rotation and scale with vectors, and then I need to ...
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64 views

Mean value of the rotation angle is 126.5°

In the paper "Applications of Quaternions to Computation with Rotations" by Eugene Salamin, 1979 (click here), they get 126.5 degrees as the mean value of the rotation angle of a random rotation (by ...
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1answer
268 views

Getting cumulative Euler angle from a single Quaternion

I've got an app that uses quaternions, and I'd like to convert each quaternion to the corresponding Euler angles. The issue is, when I convert them, the roll and yaw are bounded within 360 degrees ...
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313 views

The Quaternions and $SO(4)$

I am interested in the map $\phi:S^3 \times S^3 \to GL_4(\mathbb{R})$ given as follows: Let $(p,q) \in S^3 \times S^3$. We identify $p$ and $q$ as real quaternions with unit norms and define ...
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100 views

Update rotation matrix

Imagine you have a two noded beam in space, defined by extreme nodes 1 and 2. Image is owned by Jean-Marc Battini. To ...
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238 views

Sylow 2-Groups of a Special Linear Group

Let $SL_2(\mathbb{F}_3)$ be the special linear group over the finite field $\mathbb{F}_3$. Show that any Sylow 2-group of $SL_2(\mathbb{F}_3)$ is isomorphic to the quaternion group of order 8.
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Immersion of Quaternions

Does there exist an immersion of the Quaternion Group in the Symmetric Groups $S_6$ and $S_7$? If it does exist, can you give me an explicit description of that immersion?
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435 views

Quaternions vs Axis angle

Whats the use of representing rotation with quaternions compared to normal axis angle representation? I've been trying to learn quaternions and they make enough sense but as far as I can tell ...
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Quaternion Hermitian diagonalization

How do I go about diagonalizing such a matrix. For example let $D$ be the quaternion algebra over $\mathbb{Q}$ with $i^2 = -1, j^2 = -11$ and take the matrix: $A = \begin{pmatrix} 11 & -j-3k \\ ...
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1answer
118 views

Is there a name for a function whose square is an involution?

An involution is a function $f:X\to X$ such that $f\circ f=\text{id}$. Is there a name for a function $g:X\to X$ such that $f\equiv g\circ g$ is an involution? An example is multiplication by $\pm i$ ...
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231 views

How to understand and create quaternions?

I have to multiply two quaternions to calculate a so called spherical linear interpolation between two $R^3$ coordinate systems within the interval $t = [0, 1]$. I understand how to do the ...
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Computing a particular finite set of quaternion matrices.

Let $B = \left(\frac{-1,-11}{\mathbb{Q}}\right)$ be a choice of quaternion algebra ramifying at $11$ and consider the maximal order ...
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615 views

Is the set of quaternions $\mathbb{H}$ algebraically closed?

A skew field $K$ is said to be algebraically closed if it contains a root for every non-constant polynomial in $K[x]$. I know that this is true for $\mathbb{C}$, which is the algebraic closure of ...
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Quaternion barycentric interpolation

Let's say that i have a set of quaternions, each representing a 3-angle orientation. And with each quaternion is associated a real value (let's say a speed value for explanation's sake). Now with an ...
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1answer
53 views

limit with quaternion

Let $v\in \mathbb{H}$ and $q:t\in\mathbb{R}\rightarrow q(t)\in \mathbb{H}$. That $q(t) \neq 0$ for all $t\in \mathbb{R}$ and $q^{-1}(t) = \frac{1}{q(t)}$. So don't confuse $q^{-1}(t)$ with inverse ...
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1answer
135 views

Algorithm for finding orientation of each face on a polyhedron?

I am working on making a dice rolling application and I need to find out how far in each of the three dimensions I must rotate each of the dice to make the correct side face the camera so the user can ...
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1answer
87 views

Quaternion Matrix Multiply

My question is about applying rules of quaternions to quaternion matrices. I know that for some rotation quaternion q = [w, x, y, z], I can find the rotation of ...
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1answer
92 views

faithful irreducible representation of $A_{4} \times Q_{8}$

Construct a faithful irreducible representation of the group $A_4 \times Q_8$ $A_{4}$ is the alternating group $Q_{8}$ is the quaternions
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203 views

Quaternion group associativity

Consider the eight objects $\pm 1, \pm i, \pm j, \pm k$ with multiplication rules: $ij=k,jk=i,ki=j,ji=-k,kj=-i,ik=-j,i^2=j^2=k^2=-1$, where the minus signs behave as expected and $1$ and $-1$ ...
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218 views

Why is the dimension of $SL(2,\mathbb{H})$ equal to $15$?

Let me ask a very basic question which is inspired by reading M. Atiyah's "Geometry and physics of knots". Could you explain me (or give a reference to) the definition of the special linear group ...